There has been a lot of talk about 60 FPS like for example here (http://www.eurogamer...ch-bf4-at-60fps) where you can watch a 60 FPS BF4 video.
I watched some of it and it does look different in a way. Do you get the same results by playing a game on 60 FPS? Or this is only like this in videos?

I'm not a video specialist, but I was really interested in rendering/codecs and (technical) video editing a few years ago so I've read a lot about it.

Basically, the thing about FPS and specifically gaming videos is about motion blur. In motion picture and movies, motion blur is applied absolutely everywhere, even photos have it - that's just how cameras work, it's physically impossible to catch just one "moment" of time, if time even has a indivisible measure. This is why most attempts to increase FPS in the movie/video industry received very mixed feedback - it's hard to even notice the difference between motion blurred 30 or 60 FPS, and those who do notice it, don't really see it as an improvement in quality, they just felt like it was "faster".

Now, in computer graphics - any graphics - if you take a screenshot of something in motion, you'll just get a crisp, sharp image of that frame. This is why playing a game at 30 FPS feels like it's choppy, but watching movies or TV at 24 FPS is fine. Adding/simulating motion blur isn't hard and is a trivial task in video rendering, there were also some games and 3D engines that tried to add it on the fly, but those all failed because it just ends up feeling like input lag or "trailing".

And now the main dilemma of this whole thing when it comes to gaming videos specifically, i.e. gaming videos that feature in-game footage. Motion blur and 24 or 30 FPS are pretty much industry standard, YouTube doesn't even support higher FPS as mentioned in that article, let alone other video mass media. So, all gaming footage is also, in general, rendered to 24/30 FPS with motion blur, and we get used to that.

But watching a gaming video never feels like playing the game itself, right? I mean visually, not just because you're not playing. Motion blur is the reason for that. What 60 FPS videos make possible is completely removing motion blur without making it look choppy like with 24/30 FPS, and this, in turn, means that you can make a gaming video look MUCH closer to what the actual game looks like, and any gamer will instantly spot the difference.

This is why the FPS subject is quite popular among people who do gaming videos. It's not really important anywhere else, however, not even movie CGI effects or animation. Extra frames are going to take extra space, obviously, and allowing higher FPS will probably be very expensive for a service like YouTube just because of all the clueless people who will think higher = better (not because of the file size, but because of extra traffic and network load when people will be downloading/watching those bloated files). I doubt you will see 60 FPS on YT anytime soon.

Also, from what I can tell, that BF4 video is still motion blurred, making this entire 60 FPS thing nearly pointless. See, even tech savvy people don't really understand the whole point of it. I initially found out about this concept from KOS, a Quake moviemaker, and watching his video downloads at 50 FPS vs watching them on YT at 30FPS was a world of difference, because he rendered the 50 FPS version with no motion blur whatsoever.